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Title: Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems

Abstract

This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of freedom and spectral decomposition of reference linear operators associated with the aggregates. For rapid convergence, it is important that the resulting coarse spaces have good approximation properties. In our approach, the approximation quality can be directly improved by including more spectral degrees of freedom in the coarsening process. Further, by exploiting local coarsening and a piecewise-constant approximation when evaluating the nonlinear component, the coarse level problems are assembled and solved without ever re-visiting the fine level, an essential element for multigrid algorithms to achieve optimal scalability. Numerical examples comparing relative performance of the proposed nonlinear multigrid solvers with standard single-level approaches—Picard’s and Newton’s methods—are presented. Results show that the proposed solver consistently outperforms the single-level methods, both in efficiency and robustness.

Authors:
ORCiD logo; ; ; ;
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1670227
Alternate Identifier(s):
OSTI ID: 1671176
Report Number(s):
LLNL-JRNL-808398
Journal ID: ISSN 0045-7825; S0045782520306174; 113432; PII: S0045782520306174
Grant/Contract Number:  
AC52-07-NA27344; AC52-07NA27344
Resource Type:
Published Article
Journal Name:
Computer Methods in Applied Mechanics and Engineering
Additional Journal Information:
Journal Name: Computer Methods in Applied Mechanics and Engineering Journal Volume: 372 Journal Issue: C; Journal ID: ISSN 0045-7825
Publisher:
Elsevier
Country of Publication:
Netherlands
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Nonlinear multigrid; full approximation scheme; algebraic multigrid; local spectral coarsening; unstructured; finite volume method

Citation Formats

Lee, Chak Shing, Hamon, François, Castelletto, Nicola, Vassilevski, Panayot S., and White, Joshua A.. Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems. Netherlands: N. p., 2020. Web. https://doi.org/10.1016/j.cma.2020.113432.
Lee, Chak Shing, Hamon, François, Castelletto, Nicola, Vassilevski, Panayot S., & White, Joshua A.. Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems. Netherlands. https://doi.org/10.1016/j.cma.2020.113432
Lee, Chak Shing, Hamon, François, Castelletto, Nicola, Vassilevski, Panayot S., and White, Joshua A.. Tue . "Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems". Netherlands. https://doi.org/10.1016/j.cma.2020.113432.
@article{osti_1670227,
title = {Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems},
author = {Lee, Chak Shing and Hamon, François and Castelletto, Nicola and Vassilevski, Panayot S. and White, Joshua A.},
abstractNote = {This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of freedom and spectral decomposition of reference linear operators associated with the aggregates. For rapid convergence, it is important that the resulting coarse spaces have good approximation properties. In our approach, the approximation quality can be directly improved by including more spectral degrees of freedom in the coarsening process. Further, by exploiting local coarsening and a piecewise-constant approximation when evaluating the nonlinear component, the coarse level problems are assembled and solved without ever re-visiting the fine level, an essential element for multigrid algorithms to achieve optimal scalability. Numerical examples comparing relative performance of the proposed nonlinear multigrid solvers with standard single-level approaches—Picard’s and Newton’s methods—are presented. Results show that the proposed solver consistently outperforms the single-level methods, both in efficiency and robustness.},
doi = {10.1016/j.cma.2020.113432},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = C,
volume = 372,
place = {Netherlands},
year = {2020},
month = {12}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1016/j.cma.2020.113432

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